On deformation rings and Hecke rings
type: Item
instance of: scholarly article
title: On deformation rings and Hecke rings
zbMATH Open document ID: 0867.11032
DOI: 10.2307/2118586
published in: Annals of Mathematics. Second Series
publication date: 1997-08-10T00:00:00Z
review text: The author proves the following result: Let \(E\) be an elliptic curve defined over the rational field \(\mathbb{Q}\) with semistable reduction at the prime 3. Suppose that either \(E\) has semistable reduction at 5 or that \(\rho_{E,3}: G_{\mathbb{Q}}\to\text{GL}(2,\mathbb{F}_3)\) is absolutely irreducible when restricted to \(\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}(\sqrt{-3}))\). Then \(E\) is modular. In his fundamental paper [Ann. Math., II. Ser. 141, 443-551 (1995; Zbl 0823.11029)], \textit{A. Wiles} formulates a conjecture identifying certain Hecke rings as universal deformation rings for modular Galois representations \(\rho:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2(\overline{\mathbb{F}}_\ell)\), where \(\ell\) is an odd prime. Under some hypotheses for \(\rho\), Wiles proves that the minimal deformation ring and Hecke ring are isomorphic. The proof uses some of his results with Taylor. In the paper under review, the author extends the result of Wiles. He removes those hypotheses of Wiles on the local behaviour of \(\rho\) at the primes \(p\neq\ell\). The work of \textit{K. Ribet} [Invent. Math. 100, 431-476 (1990; Zbl 0773.11039)] on the refined conjecture of Serre, as well as an adaptation of Ribet's proof by the author [Elliptic curves, modular forms, and Fermat's last theorem, Proc. Conf. Hong Kong 1993, 22-37 (1995; Zbl 0853.11031)], are used to attach to the representation \(\rho_0\) \((\rho_0\otimes\overline{\mathbb{F}}_\ell=\rho\)) a modular form in \(S_2(\Gamma_Q)\), where \(\Gamma_Q\) is a certain group intermediate to \(\Gamma_1(N_Q)\) and \(\Gamma_0(N_Q)\), and \(N_Q\) is an integer which raises the conductor \(N_0\) of \(\rho_0\). Let \(P\) denote the set of primes \(p\equiv-1\text{ mod }\ell\) such that \(\rho_0\) restricted to the decomposition group \(D_p\) is irreducible and that \(\rho_0\) restricted to the inertia subgroup \(I_p\subseteq D_p\) reduces. In the paper of Wiles, it is assumed that \(P=\emptyset\). In the p…
reviewed by: Pilar Bayer
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 935304
zbMATH Keywords: Fermat's last theorem
zbMATH Keywords: Hecke rings as universal deformation rings for modular Galois representations
zbMATH Keywords: modular forms on quaternion algebras
Wikidata QID: Q55892256
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118586
OpenAlex ID: W2043708719
author: Fred Diamond
description: scientific article; zbMATH DE number 935304